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ThreeDimensional Brownian Motion and the Golden Ratio Rule
"... Let X = (Xt)t≥0 be a transient diffusion process in (0, ∞) with the diffusion coefficient σ> 0 and the scale function L such that Xt → ∞ as t → ∞ , let It denote its running minimum for t ≥ 0, and let θ denote the time of its ultimate minimum I ∞. Setting c(i, x) = 1−2L(x)/L(i) we show that th ..."
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Cited by 9 (4 self)
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Let X = (Xt)t≥0 be a transient diffusion process in (0, ∞) with the diffusion coefficient σ> 0 and the scale function L such that Xt → ∞ as t → ∞ , let It denote its running minimum for t ≥ 0, and let θ denote the time of its ultimate minimum I ∞. Setting c(i, x) = 1−2L(x)/L(i) we show that the stopping time τ ∗ = inf { t ≥ 0  Xt ≥ f∗(It)} minimises E(θ − τ  − θ) over all stopping times τ of X (with finite mean) where the optimal boundary f ∗ can be characterised as the minimal solution to σ 2 (f(i)) L ′ (f(i)) f ′ (i) = − c(i, f(i)) [L(f(i))−L(i)] ∫ f(i) i c ′ i (i, y) [L(y)−L(i)] σ2 (y) L ′ dy (y) staying strictly above the curve h(i) = L −1 (L(i)/2) for i> 0. In particular, when X is the radial part of threedimensional Brownian motion, we find that τ ∗ = inf t ≥ 0 ∣ Xt−It ≥ ϕ
Optimal Prediction of Resistance and Support Levels
, 2014
"... Assuming that the asset price X follows a geometric Brownian motion we study the optimal prediction problem inf 0≤τ≤T EXτ−` where the infimum is taken over stopping times τ of X and ` is a hidden aspiration level (having a potential of creating a resistance or support level for X). Adopting the ‘a ..."
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Assuming that the asset price X follows a geometric Brownian motion we study the optimal prediction problem inf 0≤τ≤T EXτ−` where the infimum is taken over stopping times τ of X and ` is a hidden aspiration level (having a potential of creating a resistance or support level for X). Adopting the ‘aspiration level hypothesis ’ and assuming that ` is independent from X we show that a wide class of admissible (nonoscillatory) laws of ` lead to unique optimal trading boundaries that can be viewed as the ‘conditional median curves ’ for the resistance and support levels (with respect to X and T). We prove the existence of these boundaries and derive the (nonlinear) integral equations which characterise them uniquely. The results are illustrated through some specific examples of admissible laws and their conditional median curves. 1.